Finite Group Databasebeta

← #31907994

Subgroups — #31907994

Order 18176.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590156C211yesmaximal, characteristic, complemented
2#22588095C211yesmaximal, characteristic, complemented
3#22586331C212yesmaximal, complemented
4#22586325C212yesmaximal, complemented
5C71×#3176C211yesmaximal, characteristic
6#17985954—21no—
7#17985820—21no—
8#17985781C2212yes—
9#17985777—21no—
10C71⋊#338C2211yescharacteristic
11#17985756C2211yescharacteristic
12#17985755—21no—
13#540×C71—21no—
14#499×C71C411yescharacteristic, complemented
15#493×C71C411yescharacteristic, complemented
16#490×C71—21no—
17#469×C71C2212yes—
18#466×C71—21no—
19#442×C71C412yescomplemented
20C4×D8×C71—21no—
21#405×C71C2211yescharacteristic
22C2×#11765849—21no—
23#14556087—21no—
24#14556068—22no—
25#14556038—22no—
26C8×Dic71—22no—
27C2×D8×C71—21no—
28C2×D8×C71C2×C412yes—
29C2×M16×C71C2311yescharacteristic
30C22×C8×C71D411yescharacteristic
31#127×C71—21no—
32#125×C71—21no—
33#123×C71—42no—
34#121×C71—22no—
35#121×C71—42no—
36C4×D4×C71—22no—
37#117×C71D411yescharacteristic
38#107×C71—21no—
39#106×C71C2×C412yes—
40#102×C71—22no—
41#102×C71—22no—
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71C2×C411yescharacteristic
45C4×C8×C71D411yescharacteristic
46#11765849—41no—
47#11765849—41no—
48C2×C71⋊C8—22no—
49C4⋊Dic71—41no—
50C22⋊Dic71—41no—
51C4×Dic71—22no—
52C71⋊C16—42no—
53C2×D4×C71—22no—
54C2×D4×C71—42no—
55C2×D4×C71—42no—
56C22×C4×C71—42no—
57C22×C4×C71C2×D411yescharacteristic
58D8×C71—42no—
59D8×C71—41no—
60D8×C71—41no—
61M16×C71—41no—
62C2×C8×C71—21no—
63C2×C8×C71—21no—
64C2×C8×C71C22⋊C411yescharacteristic
65C2×C8×C71C22⋊C411yescharacteristic
66C2×C8×C71C22×C411yescharacteristic
67C2×C8×C71C2×D411yescharacteristic
68C4⋊C4×C71—21no—
69C4⋊C4×C71—22no—
70C4⋊C4×C71—42no—
71C22⋊C4×C71—21no—
72C22⋊C4×C71—42no—
73C22⋊C4×C71—42no—
74C42×C71C22⋊C411yescharacteristic
75C42×C71C22⋊C411yescharacteristic
76C23×C71—42no—
77C23×C71AGL(1,ℤ/8)11yescharacteristic
78D4×C71—42no—
79D4×C71—42no—
80D4×C71—162no—
81D4×C71—162no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—42no—
87C2×C4×C71—162no—
88C2×C4×C71C2×C22⋊C411yescharacteristic
89C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
90C2×Dic71—42no—
91C8×C71—21no—
92C8×C71—21no—
93C8×C71—21no—
94C8×C71#10411yescharacteristic
95C8×C71#10411yescharacteristic
96C71⋊C8—42no—
97C22×C71—21no—
98C22×C71—41no—
99C22×C71—82no—
100C22×C71—82no—
101C22×C71—82no—
102C22×C71#41811yescharacteristic
103C4×C71—21no—
104C4×C71—41no—
105C4×C71—41no—
106C4×C71—82no—
107C4×C71#41011yescharacteristic
108C4×C71C2×#10411yescharacteristic
109Dic71—82no—
110#11770—711nomaximal, Sylow
111C2×C71—21no—
112C2×C71—41no—
113C2×C71—162no—
114C2×C71#150511yescharacteristic
115#3176D7111yescharacteristic
116#2166—711no—
117#1747—711no—
118#1341—712no—
119#1339—712no—
120C71#1177011yescharacteristic, complemented, Sylow
121#540—21no—
122#499Dic7111yescharacteristic, complemented
123#493Dic7111yescharacteristic, complemented
124#490—21no—
125#469C2×D7112yes—
126#466—21no—
127#442Dic7112yescomplemented
128C4×D8—21no—
129#405—1421no—
130#405C2×D7111yescharacteristic
131#364—1421no—
132#349—712no—
133#347—1421no—
134#338—711no—
135H(3,ℤ/4)—711no—
136H(3,ℤ/4)—1421no—
137C2×D8—21no—
138C2×D8C2×Dic7112yes—
139C2×M16—1421no—
140C2×M16C22×D7111yescharacteristic
141C22×C8C22⋊D7111yescharacteristic
142#127—21no—
143#125—21no—
144#123—42no—
145#121—22no—
146#121—42no—
147C4×D4—22no—
148#117—1421no—
149#117C22⋊D7111yescharacteristic
150M32—1422no—
151#107—21no—
152#106C2×Dic7112yes—
153#102—22no—
154#102—22no—
155#97—1422no—
156#97C2×Dic7111yescharacteristic
157#97C22⋊D7111yescharacteristic
158C4×C8—1422no—
159C4×C8C2×Dic7111yescharacteristic
160C4×C8C22⋊D7111yescharacteristic
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—42no—
164C22×C4—42no—
165C22×C4C2×C22⋊D7111yescharacteristic
166D8—42no—
167D8—41no—
168D8—41no—
169M16—41no—
170M16—2841no—
171M16—2841no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1422no—
175C2×C8C22⋊Dic7111yescharacteristic
176C2×C8C22⋊Dic7111yescharacteristic
177C2×C8C22×Dic7111yescharacteristic
178C2×C8C2×C22⋊D7111yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—2841no—
183C22⋊C4—21no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—2841no—
187C42—1422no—
188C42C22⋊Dic7111yescharacteristic
189C42C22⋊Dic7111yescharacteristic
190C16—2842no—
191C23—42no—
192C23#1455615411yescharacteristic
193D4—42no—
194D4—42no—
195D4—162no—
196D4—162no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—162no—
203C2×C4—2842no—
204C2×C4C2×C22⋊Dic7111yescharacteristic
205C2×C4#1455615411yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—2842no—
210C8#1455605511yescharacteristic
211C8#1455605511yescharacteristic
212C22—21no—
213C22—41no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22#1798600411yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—5682no—
223C4#1798597311yescharacteristic
224C4C2×#1455605511yescharacteristic
225C2—21no—
226C2—41no—
227C2—162no—
228C2#2258684811yescharacteristic
subgroupG/H
centerC2#22586848
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
Fitting, prime radicalC71×#3176C2
socleC2×C71#1505
cosocleC2×M16×C71C23
hypercenter#3176D71
hypocenterC71#11770
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22590156C2semidirect
2. #22588095C2semidirect
3. #22586331C2semidirect
4. #22586325C2semidirect
14. #499×C71C4semidirect
15. #493×C71C4semidirect
19. #442×C71C4semidirect
120. C71#11770semidirect
122. #499Dic71semidirect
123. #493Dic71semidirect
127. #442Dic71semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225901562
#225880952
#225863312
#225863252
C71×#31762
#1177071